## Fischer-Clifford matrices of the generalized symmetric group and some associated groups.

##### Abstract

With the classification of finite simple groups having been completed in 1981, recent work in group theory has involved the study of the structures of simple groups. The character tables of maximal subgroups of simple groups give substantive information about these groups. Most of the maximal subgroups of simple groups are of extension type. Some of the maximal subgroups of simple groups contain constituents of the generalized symmetric groups. Here we shall be interested in discussing such groups which we may call groups associated with the generalized symmetric groups. There are several well developed methods for calculating the character tables of group extensions. However Fischer [17] has given an effective method for calculating the character tables of some group extensions including the generalized symmetric group B (m, n). Actually work on the characters of wreath products with permutation groups dates back to Specht's work [61], through the works of Osima [49] and Kerber [33]. And more recently other people have worked on characters of wreath products with symmetric groups, these amongst others include Darafshesh and Iranmanesh [14], List and Mahmoud [36], Puttaswamiah [52], Read [55, 56], Saeed-Ul-Islam [59] and Stembridge [64]. It is well known that the character table of the generalized symmetric group B(m, n), where m and n are positive integers, can be constructed in GAP [22] with B(m, n) considered as the wreath product of the cyclic group Zm of order m with the symmetric group Sn' For example Pfeiffer [50] has given programmes which compute the character tables of wreath products with symmetric groups in GAP. However it may be necessary to obtain the partial character table of a group in hand rather than its complete character table. Further due to limited workspace in GAP, the wreath product method can only be used to compute character tables of B(m, n) for small values of m and n. It is for these reasons amongst others that Fischer's method is sometimes used to construct the character tables of such groups. groups B(2, 6) and B(3, 5) of orders 46080 and 29160 is done here. We have also used Programme 5.2.4 to construct the Fischer-Clifford matrices of the groups B(2, 12) and B(4, 5) of orders 222 x 35 X 52 X 7 x 11 and 213 x 3 x 5 respectively. Due to lack of space here we have given the Fischer-Clifford matrices of B(2, 12) and B(4,5) on the compact disk submitted with this thesis. However note that these matrices are the equivalent form of the Fischer-Clifford matrices of B(2, 12) and B(4,5). In [35] R.J. List has presented a method for constructing the Fischer-Clifford matrices of group extensions of an irreducible constituent of the elementary abelian group 2n by a symmetric group. The other aim of our work is to adapt the combinatorial method in [5] to the construction of the Fischer-Clifford matrices of some group extensions associated with B(m, n), using a similar method as the one used in [35]. Examples are given on the application of this adaptation to some groups associated with the groups B(2, 6), B(3,3) and B(3, 5). In this thesis we have constructed the character tables of the groups B(2, 6) and B(3,5) and some group extensions associated with these two groups and B(3, 3). We have also constructed the character tables of the groups B(2, 12) and B(4, 5) in our work, these character tables are given on the compact disk submitted with this thesis. The correctness of all the character tables constructed in this thesis has been tested in GAP. The main working programmes (Programme 2.2.3, Programme 3.1.9, Programme 3.1.10, Programme 5.2.1, Programme 5.2.4 and Programme 5.2.2) are given on the compact disk submitted with this thesis. It is anticipated that with further improvements, a number of the programmes given here will be incorporated into GAP. Indeed with further research work the programmes given here should lead to an alternative programme for computing the character table of B(m, n).